 In imam vseč na mikrostrukturale, finančne vseče in priječne in vseče. Proste, da imamo ideje, kaj je Cfm, kaj je dobro v fiziku? Cfm je zelo investičnji, zelo zelo investičnji. Kaj je zelo H fund. Kaj je investičnji? je to možno začočil za način, če je to ne možno počit, je to nebezavno. To je začočil način začin, a kaj je ahča, nega se v dete, je to reagulacija o nekaj tega inga, a ahča je nekaj regulativa, da je nekaj pension fund, je to več nekaj neč vših, The better you are, the better you are, the better you are. The better you are, the better you are. In there are some, and we are not sure about that. I think it's an interesting question. The best is the easiest question. Often we don't know, but the easiest question is In tudi nekaj dobročnih, ko se ne posavili, na začetkovne strategije, kako so četv, ko so v pomembnih odrečenih taj del, tako, da se početno početno na mali še sekund, početno početno, početno početno, in zelo je toga, kaj je vse vse skupne, kako potrebnih studij, za�ornje razdljev, kaj sem popetnem, da je tudi po vrste. To je CFM. Ono bo druga, zrč? Prejštve in v rovnih. Ga del pa CFM, tila z njivevakih tukovih svetov. Da jim mora, da se koneče se skup officiali, za razščitev, da jim se do reklami sedi. Se dela je, da se tudi koneče, in sej tudi treba in nismo pravi, da je izgleda skolid, če je inperialzia in Lundan in sej tudi je očerbega in da je zelo pozdov in P.H.G. vse, ki prišli vse. Nistak vse je zelo vse, ki je vse, ki je vse, ki je vse, kot je vse, ki je vse, ki je vse, ki je vse, ki je vse, ki je vse, ki je vse, ki je vse, ki je vse, ki je vse, kaj je čer in akonofiziks, če je... Da je niča vsega dobrovina v CfM. Nječ vse in tudi polsdok, kaj je vse, da mora. Tukaj, da se vse je objev. Tukaj, da se poživimo. Tako, nekaj je… Nekaj je, da se zelo poživimo, da nekaj nekaj. Če se je objev, in kaj je dobrovina? tak backwards here is stupid, so we could do it better. In there is a way to get in. So, as I said, there is several topics in finance, that I'll discuss. It's not at all a traditional finance or economics course because I will get into detail in a bit, so actually today it will be mostly an introduction of what are markets, what is economics, what is different that we do, but traditionally in traditional finance and economics zelo, da bomo za to včalskili počeljetni principale, če na celu da skupnega. In je ni umeličen, če za belowc, ki je časni temoček matematika, ali ne da jaz sem ne zočili. V matematikih fredž knee, je spremnik v morske, da bane kajkama matematika, ali ne za n nice matematika, to odstavljeno z prvi bilo tega všepljena in všečči pomečnji. Vo vse ci nisem bila všepljena v povesti se in z vsečnimi matematikami, ki se poverjalo na mavskih gud, nesem ne bi razliža vse za to. Zato, ki drugi sve, na razloženju pribojamo, tako da zelo, da nape torega vsej izgleduje. I pribojamo o premačno, premačno, pri prve moderale. before doing models, and then we'll try to describe this data in a proper way, and maybe unlike, I don't know how the other courses here went up to now, but I have the feeling that maybe we will be a bit less quantitative computational in this course, so we'll do calculations, we'll do math, but my idea is more is that we should understand, get an intuition of how markets work, how to model them, so the actual, if one needs to really apply these ideas, it's important to do calculations, but maybe there will be less in that sense. As I said, I will try to point out the difference between economics and economics in this course, so I will try to stop in some cases where we say, okay, that's why we are doing something different, and so the final idea is to give a new point of view of trying to understand how prices move and why they move, so instead of traditional economics approach of efficient markets, which we'll discuss, try to understand how price impact can lead to diffusive prices, so we'll understand what these notions are, but this is the goal. There will be a relatively large focus on microstructure, which I will define better, but it's again, it's because of my taste, and what I will try to do is also show some, I mean, hopefully we have time to show some applications, so to come up with models that can describe maybe prices, but also see how can we apply these to, if you want it to optimize something like trading. So there is no, okay, there is no single book that I could give you, so there are, I think there are two books which cover here and there are different parts of these, of what we discuss here, so which are not by chance, are very much related to the people I work with, so one is, one is a quite recent book of Gusho Bonah, which is titled Trades Quotes and Prices, TQP, I write it, it covers a lot of things in microstructure, so if one is interested, of course there is much more in it than what we discuss here. Another clue, sorry, so it's Trades Quotes, which is a nice title, because it's like everything. Sorry, I put TQP because sometimes, I mean, I will show some figures, I will sometimes write where I got the figure from, TQP will send for this, but you're right, so there is another book which is, which covers much less of what we discuss here, and it's much harder to read, I think, which is, again, first off, there is the same, so Gusho and Mark Potters, of which the title is Theory of Financial Risks, something. So anyway, so if you want to look into this, I mean, I don't expect you to read it at all, so we try to cover everything that we want to discuss here, but if you're interested. And so, so I want to give an outline of what we are trying to discuss, I mean, to be more concrete. So what I think we are going to do today, is it readable? Risks, this. So Theory of Financial Risks, and maybe, and derivative prices, I don't know, if you search for this, it will be, I think, you know. So the outline of the entire course is something like the following, but today we will have an introduction, OK? So what we will discuss, I think it will be a bit storytelling today, so if you want to go very deep, it's like what are markets, why do they exist, who are there, and a bit of overview of some mathematical things that we will need. Then we will have a couple of lecture on the question of prices, so what we want to look at is statistics of price moves. So what we want to discuss here is facts from, so looking at data, stylized facts, important things to know. So this is for single products, we will try to discuss the same for cool movements of prices, versus also looking at correlations between different products, try to understand how we can get information out of data, and probably get a bit to discuss how to construct a portfolio of several products if we know the correlation. And then we get two more, so this I think will be the first three lecture issues. Then we get two more microstructure related questions, so one will be, I don't know what title to give, so about what is information, what is liquidity in the market, and what is market efficiency, though efficiency we've discussed already before, but so the idea here is to understand how and why do prices move, so what causes the prices to move, try to explain how prices become diffusive, if they become diffusive on different timescales, and it will be an introduction to microstructure, so that's where we start discussing microstructure. We will have a quite long part, because due to my taste maybe on price impact, which is essentially the response of the market to what you are doing, how do prices move due to your actions, and we will have, so this will be several lectures about empirical results, first of all, of course, and try to model it on different timescales, on the micro scale, and on let's say mesoscale, so up to one day, and so this is several lectures, and probably this is also more than one lecture, and then at the end, if we have time, which I very much hope, is to get to combination of all these, something that I call optimal execution. Execution is not executing people, execution means trading, let's say, so when you actually have to go to the market and do the trades, how to do it optimally, given all the things that you've seen here, or optimally in some sense. There will be an exam, I think, and the idea is that we will cover everything to be able to do the exam, and there is this question of tutorials that might, there might be tutorials, and one other thing, so here I have some slides only with figures, but I, so I was, I think I will be able to send some notes after the lecture, so maybe after each lecture, one or two days later, some version of the notes that I have to you, people. I mean, I don't know how it works for other, is that the usual way, so do people send the notes normally? Sorry? The same day. OK, so that's also a possible approach. Yeah, OK. I don't say that today I will send about today, but today is anyway, it's a light version, I think. But OK, so then I'll try to go with that. So, OK, so, so the question is to start with, I mean, really, I think what I've been told is that it's better, since it's a subject that people didn't really study to start a simple thing, so the question is, OK, what are markets and why do we need them? So, traditionally, a market is some, OK, is a meeting place of some type of people. It depends a bit of what markets we are looking at, so one can think about, let's say, stock or bond markets, markets are typically places where, this is a traditional way to say, that there are people who have some projects, they want to do something, but they don't have financing for this, let's call them entrepreneurs, so they have project funding, let's say, so they need funding. Is it readable what I'm writing? Yes? No, there. So, OK, so I will project, but no finding, and I will try to be cleaner after it. So, this is one type of people, and there are investors, which are the opposite side of the entrepreneur. So, what's the good way? It's a bit of French word in English. So, it's people who have projects that they want to realize and don't have, what's the good way for entrepreneur? So, and there are investors who are the opposite side of the story, who have money to invest, but, and they want to gain on it, so investors have money to invest, and these people want to meet, first of all, you want to place where these two people can meet for the good for both of them, and also often, I mean, investors, so they have money and they want to invest it, but it's not obvious on what time scale, so it's, I mean, it's a triviality, what I'm saying, that you might have money you want to invest, I don't know, you have 10,000 euros, you want to invest it somewhere, but maybe two years from now you want to buy a car, so you want to be able to get your money out of this investment somehow, so, so, so you're only ready to invest your money if you think that you can exit your position easily. So, what you want is easiness exiting a position, so by position, it's what you mean, you want to be able to sell easily, and so easiness of exiting a position is often called liquidity in the market, and by easiness of exiting a position, of course, what you mean is easy to find someone to trade with, and easy to find someone to trade with at a price that can be acceptable for you, if you bought something for $100 and now another person is ready to buy it for 50 while the price doesn't really go down, you won't be happy because you want to get out easily at a proper price. And to do this, to be able to trade, so to find someone to trade with, and to have liquidity, you need markets, you need to be easily, to easily find counterparties, and what one would think is that without this system, so without investors and entrepreneurs finding each other, of course, the economy would be much slower, so this is a traditional explanation, another type of explanation between why these entrepreneurs and investors want to meet is, I will just write up the word here, is the question of risk premium, I won't go in detail, but traditionally this is a question of risk premium, and I don't know why, but I don't know why, but I don't know why, but I don't know why, but traditionally this is a bit away of understanding all movement, to explain all type of movements in the market, but it's an economist approach, it's a bit to say that there is a risk premium, so that if I'm ready to be on one side of transaction, and I'm trying to gain it, so that there is a transfer of risk, that in this case there is the entrepreneur who has a great idea and doesn't have the money, or maybe he has the money, but doesn't want to risk losing the money, the entrepreneur is ready to take a part of his risk, so give the money, say okay use my money to try to realize what you realize your gate idea, but I hope I will gain, but if not, sorry, I'm paid for taking some of the risk from you, so this is traditionally the picture about stock market, it's the idea that markets are made up so for people to meet is the same, but of course there are products which we will discuss a bit more in detail, but there are more what we call derivative products, which as you will see the definition is that the price of which the price of product is derived from another product, these are what we call derivative products, we will have a small discussion about them in ten minutes or five, but so these products also appear, so it's not this type of entrepreneur investor relation, but they behave as insurance policies, which I will explain a bit better in a second, but the idea here is that you want to ensure yourself against some large price variations, in another way you want to hedge yourself, so those people who are trading for insurance reasons are often called hedgers, so this is, and traditionally these derivative products are most important in commodity products, so oil, orange agricultural products and maybe FX rates, so we discussed a bit, the need for markets is as a meeting place for different types of traders and we discussed a bit who are the actors, so we said that there are entrepreneurs, there are investors, we said that there are hedgers and there is one other type of person which we can understand from here that we need them, but we said that we want liquidity, we need someone to make this market liquid, so if there are only these people who have great ideas or they have money to invest for a long time, they can meet at the market, but if there are not many of them, it will be hard to meet each other, so what type of extra actors you mean, so another type of actor is what we call liquidity providers, provide liquidity, who typically are, there are two groups of liquidity providers, I mean the traditional description, so there are people who explicitly, their job is to provide liquidity, we call them often market makers or who are intermediaries, so their idea is exactly, they don't want to buy or sell separately, they just want to be on both sides of the trade, if you want to sell, they buy from you and they will sell to someone else, so they are just act as intermediaries between people who have some greater idea of why they want to trade and typically we will discuss a bit in detail, of course it's not just because of goodness that they do it, they get something in exchange and there are what we call usually, well it's not a nice word, speculators, or we can call them informed traders, so who have some typically some short term information, so short term can mean different things, of course today it can mean a second, traditionally it was a longer term, but on some short term they have an information or they think that they have an information, the price of a product will go down, they want to buy it now just to get it, sell it very soon, so they don't really want to invest money on long term, they don't care about what this product really is, they just think that the price will go in one direction and trade to gain on this, so speculator is, well it's not a very nice word, but that's the case and it's a very big group of traders and traditionally it's speculators who are the most prone to hurting behaviors, so to every one of them do the same, because maybe they use the same information, or because they look at each other, they are prone to panic, so the usual idea of markets behaving in a turbulent way are often blamed on speculators, and one more mention that I want to say, so this is a very traditional way of looking at markets, if in today's markets where things are electronic, so anyone can provide liquidity, it's a bit hard to make a difference between these types of groups, so both of them provide liquidity, they are there to trade quite often, you don't know why they do, so traditionally market makers were designated, they had a sign on them that they are market makers, today you don't really know, so it's hard to make a difference. So this is what are markets and why do we have them, I wanted to give a short discussion of what the different products that are traded in, that we will discuss what they are, it's partly for general culture, of course I mean one wants to know when we are discussing, in the following, when we will discuss data, you want to know what these products are, but it's also for general culture that we want to look at this. So we most often look at stocks, so stocks or you can call them, I mean shares, I think this is probably an ownership in a company, transactional ownership of course and there are public companies of which anyone can get stocks or get a partial ownership, I don't know if you go to the stock market you can buy stocks of Apple easily, these are public companies and there are private companies, sorry private stocks, of which stocks can be bought only directly from the owner, so not on the stock market. We will of course mostly think about public companies in the following. Having an ownership of a company means that you really own it so you have rights to vote in the decisions and different stocks can go with different voting rights, we don't care about these details I think one important thing for general culture of course is that if a company makes profit, then this profit can either be a company or can be paid out to the owners of the shares as dividend, this is important for general culture and it's also important that of course these cos predictable changes in the price, if you know that the fact of owning a stock on a given day will give you the right dividend that of course has some mechanical effects on the price of this stock. One important thing that often we do not think about is that in case of stocks but in other cases as well but in case of stock there are several things that can happen to a company, a company can go bankrupt and company can be taken over by another one so there are several changes that can, especially when we are studying data can give bias so if you only study stocks that were available in your data set from 1995 to 2005 on the first day on the last day they were both there it's a huge bias in your studies because you don't take into account all of them that went bankrupt in this period so it's just to be careful. We also often discuss stock indices which is essentially a stock index is a basket of stocks so the price of a stock index is some type of average of the prices of the stocks in this basket it can be it can be some different of course it can be a flat average it can be weighted by the size of the company in some definition and it can be defined for different categories so a given sector can have an index it's not very important and especially in case of stock indices here it's important this type of selection bias and survival bias of course a stock index can change its constituents so the basket can change in time so you have to be careful when do a study what I discussed here is I mentioned already this derivative product so futures contracts futures or forward contracts a third group that we will often discuss which is the derivative product this price is derived from another so futures is this type of contract the idea is that you buy or sell an asset today to be delivered sometime in the future so you pay the price that is defined today and you will get in the future so buy sell today the future so the idea here is exactly that's why I said that these are insurance products you can buy a futures contract today let's say you know that you will need crude oil in the summer but you're afraid that the price of it will go up until then so you can buy a futures contract now to be delivered in the summer so you pay the price now you will get the crude oil in the summer so you're insured against the price going up like crazy if today the price is 100 in the summer it will be 150, you will be very happy that you bought the futures contract today of course if the price went down bad things can happen then you lost on the deal but you hedge yourself, you insure yourself against large price changes they exist, no so not for any market they exist for for many products but they are mostly liquid on so on indices they are very liquid on agriculture and commodity products I said because there it's really if you produce something it can be important for an exchange product so USD, GBP rate or something like this in many markets if I change my mind sure, ok so what we discussed here are all traded on on existing markets so if you find someone to sell it to no problem of selling it so ok, actually I want to say one more actually the different I wrote futures and forward which is the same idea the difference is that forward is what we call over the counter product so you buy it from someone, I buy it from you so you will have to deliver I mean you are delivering me later it's a one-to-one match you have to convince if you want to get out of the deal you have to convince me that I'm ok to trade with him so it's not a centralized market the futures version of the contract is traded on a market on which you buy from someone you don't even see the person it's an invisible person to you there is a marketing between you so sorry to answer the question, yes you can get out of the deal of course but what it means that you have to convince someone so if you bought a to get crude oil in the summer you bought it for 100 and now you see that the order and the price went down now it's 80 so you say oh I want to get out because I could do better you still have to convince someone to buy this from you so if it has no value anymore because the price is very far from what current price from what you paid it will be hard to convince someone to get it from you I don't know if it's clear what I'm so these are futures contracts and another type of product I want to discuss are options which are also derivative products but they are a bit more complicated than futures I want to spend a minute on these so what we said here so futures is buy now to get delivered later in the future but it's a contract so I mean it's if the price went in the wrong direction you will be sad there are options which are again they are derivative products of which we see from the names there is an optionality which means that it is the right but not the obligation to buy in the future at a later date for the price that you decided today so it's while these futures are obligation this is a right not obligation to buy something at a later date for the price that you decided today so in that sense it's like a futures contract yes so the word derivative simply comes that the price of this depends on the price of something else so what this happens so let's give an example here you could say that you buy an option let's say on a stock so for simplicity on apple stock you buy an option that will have a that has a I will do another way I will draw up something and I think that will clarify the question so just to say so this is right but not obligation to buy sell at a later expiry and get delivered later expiry date there are two type of options that there are call options and put options it's simply a jargon so call is to buy the product and put is to sell and and so I will draw up what they call the payoff function of an option and I think that will explain the question so the payoff function essentially what one looks at in a case let's say if you buy a call option you can write up the profit that you are going to make as a function of the price of the product at expiry of the sorry of the underlying product at expiry so because this derivative product you buy the option to get delivered something later and this something is the underlying this is called the underlying so what we what we want to plot here is the profit as a function of the price of the underlying at expiry so if you bought a call option you pay the price for it which is there is a strike price in this option so the price at which you will get this to exercise this option at expiry this is called strike price so what does this option worth if the price at expiry is below the strike price is here so the strike is here if the price of the option at expiry is below this price means that you have the option to buy apple for 100 euros in a month and then you arrive to that moment in a month and apple is worth 80 then it is not worth anything you won't exercise this option you will throw it out in the bin so you made no profit if the price below this strike price while if it's above in the same way you start gaining if you have the right to buy it for 100 and the current price is 105 you gained 5 euros so if the price at expiry is above the strike price you simply gain the difference between the price and the strike price so from this we already see that the price of this product and the profit on this option product depends on the price of the expiry so it is derived from the price of the underlying and in practice so this is the ideal picture of actually I have a figure about this so this is the ideal picture of your profit but in practice of course when you got the option you had to pay out something you didn't get it for free so you had to pay something which is called the premium and that has to be subtracted from this curve so it will be somehow like this the actual I don't know if you have colors so your actual profit will be something like this if the option is useless at the end you lost what you paid when you got this option and if you start gaining on this still you have to stop track this premium that you paid so this is what we see this is what we see here so long call that being long means buying short means selling buying a call and long call is the same so these are options market we don't discuss them in details later but I think it's important to know we will mention them sometimes of course here we have for put options selling a call option once one understands one of these curves the others are very simple if you also bought the product so what I'm talking about here is that you buy the right to get Apple stocks for 100 in end of March if you arrive end of March and the price of Apple is 80 then this is useless for you but you're saying if I understand well that at the same time if you have stocks of Apple in your pocket or if at the same time you know you have to buy if you have Apple stocks in your pocket so you have something else as well not just this option that can change the entire payoff but only having these options in your pocket if the price now in the market is 80 and I have the right to buy it for 100 that doesn't help me I can go and buy it for 80 but that has nothing to do with this option did I answer or not so don't forget it so these will be a couple of types of products that we will discuss in the future and we discussed a bit why these markets exist so I wanted to go a bit into discussing what is economics versus economics which I mentioned several times so the crucial difference is the crucial difference in between economics and economics is the role of concepts, the role of equations and the role of data so the crucial difference between economics and economics is the role of these three things so concepts, equations and data it seems to be very similar economics versus economics and what is the main thing is that in classical economics there are very strong assumptions or sort of axioms if you think about mathematics that are assumed and might not be true so in classical economics typically you have three big axioms one is which is called rationality one is which is often called invisible hand and one which we will discuss more in detail so in case just to make sure I'm explaining this here to make clear what the difference is I don't expect you to learn the definitions of invisible hand it's more just to get an understanding if it's not the type of question you ask so what is rationality? it's easy to understand the word but typically I looked at Wikipedia how they define the rational agent and it says an agent that has clear preferences models uncertainty via expected values of variables or functions of variables and always chooses to perform the action with the optimal expected outcome from among all feasible possibilities so what one immediately feels from this is that it's something very idealized sure you might have clear preferences you try to model uncertainty in some way but it becomes very hard but especially this idea of choose to perform the best actions among all of them but I think we know that it's never the case you're never doing this at most you look at the subset of of the possibilities so this is the definition of rational agents this is the way economics looks at actors in the market and the way if everyone is rational somehow in the same way having the same type of information then actually what economics says is that you can instead of having all these agents you can just model one representative agent because they are all the same they have the same information they are optimizing the same things if there is no difference between them you can simplify modeling I guess it's clear that this is quite far from reality another thing which is invisible hand which is this is a very important economist who says that each individual in pursuing his own interest or his own selfish good is led as by an invisible hand so that's why it's called invisible hand is led to achieve the best good for all of society so what it says that we are all trying to optimize something for ourselves and this will bring global optimum for society and any interference any interference into free competition was is almost certainly injurious is bad for society so ok this this is what's invisible hand and efficiency which I think is more interesting is says that prices reflect all information public or private and no one can learn access returns so prices have all information in them you cannot just go to the market you cannot have extra information to earn money so we know that this is not true there are people at least who try to earn money and for long periods can earn money but somehow we have the feeling that so there are these three axioms of economics about some things you have the feeling that it's not true from a part you might not have the feeling it's not true or not it's not true or disproved so somehow you have the feeling from these three that it's a bit of a political propaganda so that markets will go to the optimum of society bring the society to its optimum and and and find the perfect prices so so there are these type of axioms in economics but the most important thing is in economics axioms are stronger than observations so these are said and there are several papers of economics one can look at we say that these are stronger than facts these are the truth of the world and if data seems to show something different then that's the problem of the data so that's why physicists arrived who think that observations matter and usually physicists say that if a model is not compatible with the data then they try to throw away the model not the data and in economics if data and models do not match then they call it an anomaly that they don't try to model so this is the type of type of problem and just a remark that what one could say is that there are a lot of physicists working in banks since the 80s since the end of the Cold War there are many physicists who went into banking where the physics approach to look at data before looking at axioms and apart from not being true of course this type of oversimplified view of the world is super dangerous for example if you want to estimate a distribution so as we have said here in rational agents are very good to estimate expected values but to estimate a distribution is very tough so from real data and that is of course related to the probability of distribution in the market and estimating for example joint probability so of several random variables the joint probability is even harder of course and if we don't estimate them well that's when we get an entire simultaneous default of several products and just to give some examples of this so we'll see examples I think in the later on I won't go into the data but the new approach is to this so this is the traditional approach in economics there are new approaches one from inside economics which is called behavior of finance which understands that people are not so rational or not rational in the same way and the econophysics on the other side and I just wanted to put a quotation from Carl Soms who is a super very famous economist and he seems to be much more optimist than me so this one in 2002 but he says that what he hoped is that in the past two decades so this was almost 20 years ago economics has witnessed an important paradigmatic change paradigmatic change with at least three closely related expects so from representative agent going to heterogeneous agent system so from this get around saying that people are not the same from a full rationality to go to some bounded rationality that there are limits to how rational you can be and getting from something which is mainly analytical which is sorry I didn't mention this but this is a big problem in the idea that economics is axiomatic that economists want to have models that can be treated analytically economists want to be mathematicians and if you look in an economics paper you have theorems in it and so they have models that are analytically tractable and instead the idea is to get to more computational approach which is analytically great but it's often not the case and similarly in finance he claims a similar shift seems to occur so from a complete rationality getting to bounded rationality and understanding that prices can be driven by something else than this rationality by the psychology of the actors in the market so so yes this is a bit the difference between in a very quick view but I don't think it's useful but it's used it remains a dog so I have the feeling that there are several I might not be the right person to discuss this because of course I'm on one side of this picture but there are two communities of traditional economics there is behavioral finance and that kind of physics which is much smaller I'm discussing this here and I will discuss more but real economists do not consider behavioral finance and then there is game theory some theory in between but there is not that much discussion between so it's I think everything falsifies in the visible hand but so just one last thing that I wanted to can be so I'm not sure I understand when there is a bounded rationality and when the person is fully rational then he knows all the consequences of course he won't go for a deal which has some laws associated with it bounded rationality doesn't mean that you know the future the probabilities you buy something and you are perfectly able to estimate the probability of having a loss more than 10% of course the problem is this is that if you look at how people behave they are very bad at estimating probabilities I mean it's easy to estimate the probability of one half but it's very easy very difficult to make a difference between probability of 10 to the minus 3 or 10 to the minus 8 so it's more bounded rationality in this sense means that you understand that you cannot for one example but you understand that you cannot estimate a probability so you say that the maximum amount that you buy is limited by the fact that you say that I cannot estimate I cannot say that it's zero probability to lose everything of it so I want to buy something that even if I lose it I can live afterwards it is a super simple example bounded rationality can be defined in some way why does it grow for the bounded rationality the probability distribution is there just you are not able to estimate it in this example exactly so for example can be that you understand that the variance that you measured might not be the good measure might not exist on your data you can always measure variance but maybe the real distribution does not have finite variance it can be the case so I wanted to be a bit related to this just before we get to a bit more concrete things I just wanted to put up this quotation which is from just after the crisis in 2009 Emmanuel Derman and Paul Wilmot both of them background in physics mathematics but working in finance who wrote this financial model as manifesto which is a bit paraphrasing because it is not a model and at the end there is this modeling which is I mean one can think about this after having done a course but I think it is good to look at it so the idea is of course it is against all this approach of traditional finance so I remember that I did not make the world and it does not satisfy my equations I don't think I have to explain this though I do use models because I won't be overly impressed by mathematics so it's not the goal I would never sacrifice reality for elegance without at least explaining why I have done so so it can be the case that you want to sacrifice but you want to be clear I think this is very very important so I won't give people who use my model this false comfort of its accuracy and instead I will try to make explicit assumptions and oversight so of course you are doing models you are not able to describe everything but the main danger of traditional economics models is that it's often hidden under the rug the assumptions and of course that's the last point it's just in general you understand that your work may have enormous effects on society these are important things to understand in the model and many of them beyond our comprehension so why do we this is the difference a bit about economics and we can get to more concrete points in a second why do we want to understand markets because you could say why is it interesting I think it's an intellectual challenge somehow if you come from physics you understand that market is somehow a strongly interacting system with feedbacks with behavioral biases of agents and of course for practical reasons if you want to trade of course you want to limit your risk you want to protect yourself and have a better understanding but one other very important thing and why our work and some of this I will discuss is important is for regulation of course the stability of markets is important also for those who do not trade in the market explicitly I mean if there is a crisis it has an effect of all of us so to have proper regulation is very important so this was sort of the introduction this and I wanted to a bit discuss what so what type of data we will be looking at so we will be discussing time series often you have seen them in most of this most of the lectures it will be time series that are most probably probably one dimensional so like prices of course this is the big example or it could be any other time series I mean anyone can come to the time series example the position of a random walker we will discuss often or the temperature in grinyano or something like in time and we will have two assumptions very often which are usually not true so we have to keep in mind and sometimes handle data to make it true or to consider so one is stationarity that we will have which is I guess you know what it is we will discuss very simple things now but it is the fact that the change so the change on a time scale to of a process which so simply define as x of t plus tau minus x of t is independent of t so it is some type of time translation in variance that you consider about data it is often not true so I mean one has to be careful data is often not stationary and the other thing which is related to this is what you often assume is that there are no overall trends which is essentially related to this so what you say is that the expectation of this over time is zero it is often not the case actually very clearly if one looks at prices in the last 50 years stock market prices are typically going up on average so often to handle this depends on what is interested in but often what you care about is the fluctuations of the process around so you might want to detrend the data if the trend is there overall you might want to do some detrending to be able to look at the fluctuations of of the movements around the mean you might also want to do a detrending that depends on the period if there are periods when the trend is upwards periods when it is downwards so one has to be careful but these two assumptions can help in a lot of analysis then you can message the data a bit to make them true without destroying information sorry no, there is no explicit dependence on tau here so you can we are talking about the average here so even if they are far away what you want, it might not be true for data and then you want to handle it what you want is that there is no clear trend in it so it is not that for any tau you don't want this to be different from zero and if it is different from zero you might want to do some detrending or maybe that is the information for you it depends if you are interested in the fluctuation or the trend itself but mostly we assume these it is a time average it is on one long data set I mean in this sense in data you are averaging in time if this question of ensemble average against time average is a bit difficult I mean it could be the case that you have several different products and you want to we have, which have somehow the same dynamics and then you can think about an ensemble average so but it is a time so but still very often we are no, it is not correlated so I mean this is a no, but it doesn't mean that there isn't any correlation in the system it just means on average the change in a given window now and later doesn't depend on time yes, so what I mean here is the difference between two so you can have a work which does some process which does this I don't know, something going down so of course you have local fluctuations in it but overall you have a trend so if you set I mean this is, I don't know if this length is, I don't know 50 setting tau equal to 50 you are averaging all the different things, you have fluctuations but in general it's an upward trending data right? this difference so this is x of t so you look at the difference between here and I don't know here it's upwards, of course there are examples where the difference between here and here is downwards, but on average there is a trend above the fluctuations, I mean it's a sum of fluctuations on trends while you can have a process which something similar, but like this I mean you can take off you can try to fit the trend in it you can try to say, somehow it's this you can have some statistical method take it off, I'm not sure if I'm maybe I'm not explaining what you are asking here we are x t so we are discussing in general so x t can be anything here it's a stochastic process yes we are making the this is far from reality indeed it can happen I mean it doesn't make the question simple the fact that there is no trend in it because typically the trend is the fluctuations are more hard to understand than the trend for me and in generalizing indeed it can be not true I mean in general that's what I said of course I write x here but x can be a price and prices on long times might be going up the economy is growing and on average prices are going up so indeed it can not really small, but if you look at actual prices from 1950s to fluctuations are not small in reality, but especially it's the fluctuations that you are interested in of course if you are able to measure this I mean it depends on what amount of data you have because fluctuations you are averaging out and the trend you are not averaging out so the ratio of trend versus fluctuations comes into the picture as the length of the data if you have 10,000 years of data of course you are able to measure easier even if it's a small trend in it it's easier to maybe we can discuss it but in general still you care about about how the signal I lost my normally what you care is how the signal changes on some timescale but if you have detrended data of course then this type of measure doesn't help you so typically what one wants to look at is somehow the measure of the standard deviation so so often what you care about is more the standard deviation of data which is the square root of the variance so the variance which is simply defined as the expectation of the square of this stuff so this is what you call actually variogram we will discuss this sometimes so the variogram which is dependence of this quantity on the timescale tau so this we will use quite often we will look at it so just to give examples of this so this is a V some examples of how the variogram looks like so one can look of course if you have the xi point iid there is no they are independent then you will measure v tau which is independent of tau which is a trivial and not really interesting case but you can look at of course brownian motion in which case I won't write this up, show this but I think you looked at it standard no sorry I didn't get this is v, okay sorry I don't know if it's this but actually I mean it's the second line which is interesting so it's just one can think about the standard deviation of the variance so one is the square of this okay sorry so it wasn't visible okay so in case of a brownian motion which I think you looked at it in the other courses is v tau is linear growing in time scale with some diffusion diffusion constant so this is what we call a normal diffusion and one can think about anomalous diffusion point which is a generalization this is visible from here now it's visible so anomalous diffusion which is a generalization so that you can say that this varial gram grows as h exponent where h is what we call hrst exponent so usually this is the general point of how the variance scales, this is called the hrst exponent and in general one can write up that one can try to plot the variance as a function of time scale where you can have something where this is the normal diffusion so which is h equal to one half so this would be an exponent one you can have h below one half which is a mere reverting process that's what sub diffusive let's call it sub diffusive but in the next in finance it's more like mere reverting that one uses above zero five which is super diffusive so this is a measure that we often look at and just to get an example one further example is one can think about one type of mere reverting process which is actually often seen in case of stochastic processes where you say that your process behaves like this k plus one is something like this I'm trying to write properly but I'm not able so you can define a process like this where this eta is some normally distributed so with some sigma squared so these are Gaussian distributed steps and of course so this is some type of mere reverting process you see that that tries to bring the process back to the origin so if you set omega equals zero then this is a simple random walk without mere reverting then this is just one and then it's easy so the position at x files one is the position x k plus the step, the Gaussian step and if one wants to write up the variance of this process first of all, okay you can write up the same, it's just rewriting this you can write up so one can write up essentially from this process just write up essentially a geometric series for the position at time k of this walker and I won't go very much in detail here but this is quite easy, one can quite easily write up, so this is a mere reverting process if you look at k going to infinity so for long times so that the initial so that the process is stationary you can write up the variogram of this process essentially writing up a sum of a geometric series and taking the square you get something like this I think so this is what you say get for this quantity for the variogram as a function of timescale which means that if one wants to plot this what does it mean for short times you can do some series expansion and you will get a linear term with a certain slope so if you plot veto as a function of the timescale what you get is you get some linear increase in the beginning and for large times of course this guy will disappear so large tau this guy will disappear you are going to a constant which is sigma square over omega so for large times you are somehow going here well in between one can calculate if you don't do it now so for example this is an interesting process because what you see is that for long times you saturate at this point the variance saturates but for short times you have some normal diffusion you don't feel the strength of the spring so this is normal diffusion why I wrote up this toy this toy here it's a well known model in stochastic processes it's called a process and it can be used in physics it's in some big particles in some vicious system it's used in finance for interesting models we won't really use it but I wanted to show it I mean it will come up but I wanted to show it an example of calculating the variogram for a mirror burning process one very related thing that we'll often look at is what we call signature plot which is essentially the same as the variogram divided by the time scale on which it has been calculated it doesn't add more information but visually it can help us understanding how the data looks like for some reason don't ask me why it's called signature plot which is what we've shown before variogram at time scale tau divided by tau this is what we usually as a function of tau this is what we call signature plot I will show you in a second and I will show I just want to do a quick calculation about how this signature plot should look like for some very idealized price process so let's assume that you have a price which is the following some reference price at time zero plus all the price changes since then zero to t minus one so all the price now is the price before plus all the price changes and just for simplicity I will write it like this so the scale of this is put in p hat it's some typical scale of the price it's a constant here very far and then so this is of the order of magnitude of r doesn't contain p anymore we took it out so what you think you can say that the f you hope that that this guy here has zero mean and some some variance this is an r here and and let's further assume that there is some correlation between the steps here which might not be a very financial approximation but for this model we can say that the correlation ok so we are here doing a problem because I will discuss a bit of a correlation exactly on that slide on the other side of the board in one minute but you know what a correlation is so we can use it before defining it's also let's say that the correlation simply is define it like this or correlation so of course for example if this was a simple random walk so there is no correlation then one could say that cr of tau is of zero and tau so if you want to write up this measure that we had here so this what we call signature plot you need to define so in this process of course one can write up the variable gram so what is it it's this expectation and and one can write this out properly so this would be of course which is I'm just writing out the definition which can be of course written up as the following double sum can be written somehow like this so what I'm doing here is super simple I'm getting this definition and this definition for the variable gram I'm putting it together so the variance at times square tau will be the square of this in which you can write up as two sums on t and t prime or t prime and t double prime and put the expectation inside some of expectations what's the letter sorry yeah so it's what we so what we say here that you have a price process p the price at time t it's a very simple model price at time t is some initial price at time zero plus all the price changes since then simply you say non-overlapping window I will get the price today so those are r which is the steps of this process and for simplicity I take this p hat out essentially I take the scale of it out so one could define this instead of of having an r till the here without the pi without the p hat and I just take a scale so this is a constant number that I put in front so r is the steps of this I mean this is the definition of the process I can define a process like this and then I can give an interpretation that this is a price and these are returns that's why it's r but this is in itself a definition of process but it can be if you think about the price it's the change of a price in a big window is a sum of the changes in small windows no I mean p, I know a pt and of course I'm averaging over time pt so what we had in a blackboard that I cancelled so you're looking at the change of this signal in a time with the tau look at the square of this because you think that there is no overall trend so it's the second moment that can be interesting for you and then you take the average of this is it so so to get on so one can write up this sum here right here below it's completely non visible right so of course this sum that we have here is well these are the definitions of correlations essentially so of course one can write up so continuing this this will be something like so you have a sigma r square because of the definition of correlation this way sorry I made an error here so of course this is p hat so you will have something like this ok so this is trivial and you have different terms of the correlation that you're summing here you're summing on the two variables so well I can write it out but it could be homework so you have tau cases when the lag between t prime and t double prime is zero so tau times c at lag zero plus one can count the different terms one can write this up so it won't finish them but it's easy to write up which at the end if you if you want to write up this initial guy here so v tau over tau so now verified v tau it will be so sigma r square p hat squared and this will be all the sum if one runs right up sum here so this is a u so I'm doing a really basic algebra here so one can write up this sum in the following way and why is it interesting what we have here now we can think a bit about the process so ok what do you see so this variogram divided by the time scale which we call a signature plot we'll have some constant plus this sum behavior and actually one can play around so ok what does this mean first of all these are positive terms you're summing positive terms so if the correlation is positive then then you will have the variance increases this time so it will be constant plus something if it's negative then the variance decreases this time scale so you're summing more and more negative numbers here because these are zero if the leg is zero the correlation is one I mean it can have the times if it's defined like this it's a correlation at zero of the same process in practice it's actually the variance of the process that's why we normalize by the variance here to have this go to one so what we see here is that for positive correlations this v will be increasing we start with negative so we have something like this type of behavior for a simple world we can say that the correlation is somehow decreasing exponentially so it's some row correlation number to the power u we can define a correlation decreasing correlation then one can define if row is zero so if there is no correlation of course this signature plot v to over to will be flat because then this is summing zero and it's a number instead if the correlation is positive you will have some increasing behavior up to some point up to the point where the correlation decays to zero if it decays to zero normally it does similarly I don't know why so this is one case if there is uncorrelated if it's positively correlated you expect something like this and if it's negatively correlated you expect some decreasing behavior initially because you're summing negative numbers here and probably it goes to some for long time it will flatten out because all correlations die out when does how much time do we have no, I had in mind that it will be much less so I have to get used to measuring time properly so this is a way so what do we do here the signature plot and the variogram is one way to look at how the data is correlated if we look at the type of figure like this if this v to over to is decreasing it's an anti-correlating process sequential steps are anti-correlated if it's flat that's the most interesting it's a normal diffusion and it's often more interesting I mean one could of course look at correlations themselves visually so I just want to say a quick word so why don't we just look at correlations of course look at correlations but sometimes in time series it's good to be careful I should have written what I'm doing so if it's this then you say that if this guy is decreasing it's a negative correlation because what happens is that you have a constant plus something that is summing up negative numbers and then of course this guy would be positive correlations and in this case we have exponentially decaying so they flatten out but for another correlation they could be different so one could look at correlations directly so one could say that instead of signature plots and variograms why don't we look at correlations and so one can write up one way to define correlations usually I mean it's always a question if you are centering the correlation if you are normalizing it or not but if you write the correlation in the following manner so you are centering your correlation then one can in homework but it's very trivial to write up that what we had before this variogram for time like tau can be written simply up simply as 2 times correlation at zero minus correlation at tau which is essentially very simply taking the variogram writing it up it will so it's a one to one correspondence so we could say why don't we look at correlation why do we define new type of variables in practice correlations are often not very useful they can be slightly misleading I just want to give an example without going through it so the problem is when correlations times are very decay very long times they change correlations if one wants to write up an empirical correlation I just want to give this example if you have an empirical correlation that usually you have your data X and you define it in the following way you say C empirical for a leg L is the following thing so something like this where what I define here so you have initially an X I process usually so this X hat is simply defined as the original process minus its mean so which is simply XK minus X mean so it's a trivial thing what you're doing is you have a process you have real data you take of the mean normal age because that's the way you define your correlation here it's not zero mean in itself and you calculate the empirical correlation but you say you go with windows for a window of size L you will have N minus L points on all of these you can sum up what this product gives and you normalize it so it's very simple no problem is the following is that that one can show that exactly the following equality exactly holds so it's with unfortunately is it visible? sorry yeah is this visible what I wrote here? ok so what I just say I'm speeding up sorry so that if this is your definition of your correlation one can write up this exact equation so you're summing L from zero to N one minus L over N actually it's very similar shape by chance see empirical like L this sums to zero one can write up that well ok one can write up very simply the different terms the problem here is that if you have a process in which the actual real underlying correlation is long range goes long range and is always positive then of course you can never have this sum go to zero so if the real correlation is always positive and doesn't become zero very fast so ok in practice if you have a correlation where you have the real corral time the characteristic correlation is L zero where somehow this L zero is much larger than one of course it's less than N I mean you don't know what happens after that so if the correlation is always positive this cannot hold so in practice your empirical correlation will have to have some purest negative terms what you will measure in the following function of L you measure the correlation if the true correlation is something like this for this equality to hold you will have to measure your empirical correlation will follow this hopefully but somehow it will have some purest negative terms for this sum to go to zero be careful when you're looking at actual date your correlations might not be good good estimations of the underlying of what's underlying ok so so what's the rule normally here I have some more things to say should we do it tomorrow and should I cut stuff or should I finish it I don't know but what's the normal rule tomorrow ok so ok just a second is there ok ok, so let's continue tomorrow sorry